In: Advanced Math
Direct product of groups: Let (G, ∗G) and (H, ∗H) be groups, with identity elements eG and eH, respectively. Let g be any element of G, and h any element of H. (a) Show that the set G × H has a natural group structure under the operation (∗G, ∗H). What is the identity element of G × H with this structure? What is the inverse of the element (g, h) ∈ G × H? (b) Show that the map iG : G → G × H given by iG(g) = (g, eH) is a group homomorphism. Is it injective? Surjective? Do the same for the map iH : H → G × H given by iH(h) = (eG, h). (c) Show that the map πG : G × H → G given by πG (g, h) = g is a group homomorphism. Is it injective? Surjective? Do the same for the map πH : G × H → H given by πH (g, h) = h. (d) Prove that the image of iG is the kernel of πH, and that the image of iH is the kernel of πG